Difference between revisions of "2017 AMC 10B Problems/Problem 8"

Revision as of 15:33, 5 February 2024

Problem

Points $A(11, 9)$ and $B(2, -3)$ are vertices of $\triangle ABC$ with $AB=AC$ . The altitude from $A$ meets the opposite side at $D(-1, 3)$ . What are the coordinates of point $C$ ?

$\textbf{(A)}\ (-8, 9)\qquad\textbf{(B)}\ (-4, 8)\qquad\textbf{(C)}\ (-4, 9)\qquad\textbf{(D)}\ (-2, 3)\qquad\textbf{(E)}\ (-1, 0)$

Solution 1

Since $AB = AC$ , then $\triangle ABC$ is isosceles, so $BD = CD$ . Therefore, the coordinates of $C$ are $(-1 - 3, 3 + 6) = \boxed{\textbf{(C) } (-4,9)}$ .

$[asy] pair A,B,C,D; A=(11,9); B=(2,-3); C=(-4,9); D=(-1,3); draw(A--B--C--cycle); draw(A--D); draw(rightanglemark(A,D,B)); label("$A$",A,E); label("$B$",B,S); label("$D$",D,W); label("$C$",C,N); [/asy]$

Solution 2

Calculating the equation of the line running between points $B$ and $D$ , $y = -2x + 1$ . The only coordinate of $C$ that is also on this line is $\boxed{\textbf{(C) } (-4,9)}$ .

Solution 3

Similar to the first solution, because the triangle is isosceles, then the line drawn in the middle separates the triangle into two smaller congruent triangles. To get from $B$ to $D$ , we go to the right $3$ and up $6$ . Then to get to point $C$ from point $D$ , we go to the right $3$ and up $6$ , getting us the coordinates $\boxed{\textbf{(C) } (-4,9)}$ . ~ $\text{KLBBC}$

Solution 4

As stated in solution 1, the triangle is isosceles.

$[asy] pair A,B,C,D; A=(11,9); B=(2,-3); C=(-4,9); D=(-1,3); draw(A--B--C--cycle); draw(A--D); draw(rightanglemark(A,D,B)); label("$A$",A,E); label("$B$",B,S); label("$D$",D,W); label("$C$",C,N); [/asy]$

This means that $D(-1, 3)$ is the midpoint of $B(2, -3)$ and $C(x,y)$ . So $\frac{x+2}{2}$ $= -1$ and so $x = -4$ . Similarly for $y$ , we have $\frac{y-3}{2}$ $=3$ and so $y=9$ . So our final answer is $\boxed{\textbf{(C) } (-4,9)}$ .

- youtube.com/indianmathguy

Video Solution

https://youtu.be/4rRckA3gcPU

~savannahsolver

Video Solution by TheBeautyofMath

https://youtu.be/XRfOULUmWbY?t=367

~IceMatrix

@@ Line 49: / Line 49: @@
 This means that <math>D(-1, 3)</math> is the midpoint of <math>B(2, -3)</math> and <math>C(x,y)</math>. So <math>\frac{x+2}{2}</math> <math>= -1</math> and so <math>x = -4</math>. Similarly for <math>y</math>, we have <math>\frac{y-3}{2}</math> <math>=3</math> and so <math>y=9</math>. So our final answer is <math>\boxed{\textbf{(C) } (-4,9)}</math>.
-- abed_nadir
+- youtube.com/indianmathguy
 ==Video Solution==

2017 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 7	Followed by Problem 9
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

Page

Toolbox

Search